Orbital period

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The orbital period is the time taken for a given object to make one complete orbit about another object.

When mentioned without further qualification in astronomy this refers to the sidereal period of an astronomical object, which is calculated with respect to the stars.

There are several kinds of orbital periods for objects around the Sun:

• The sidereal period is the time that it takes the object to make one full orbit around the sun, relative to the stars. This is considered to be an object's true orbital period.
• The synodic period is the time that it takes for the object to reappear at the same point in the sky, relative to the Sun, as observed from Earth; i.e. returns to the same elongation (and planetary phase). This is the time that elapses between two successive conjunctions with the Sun and is the object's Earth-apparent orbital period. The synodic period differs from the sidereal period since Earth itself revolves around the Sun.
• The draconitic period is the time that elapses between two passages of the object at its ascending node, the point of its orbit where it crosses the ecliptic from the southern to the northern hemisphere. It differs from the sidereal period because the object's line of nodes typically precesses or recesses slowly.
• The anomalistic period is the time that elapses between two passages of the object at its perihelion, the point of its closest approach to the Sun. It differs from the sidereal period because the object's semimajor axis typically advances slowly.
• The tropical period, finally, is the time that elapses between two passages of the object at right ascension zero. It is slightly shorter than the sidereal period because the vernal point precesses.

## Relation between sidereal and synodic period

Copernicus devised a mathematical formula to calculate a planet's sidereal period from its synodic period.

Using the abbreviations

E = the sidereal period of Earth (a sidereal year, not the same as a tropical year)
P = the sidereal period of the other planet
S = the synodic period of the other planet (as seen from Earth)

During the time S, the Earth moves over an angle of (360°/E)S (assuming a circular orbit) and the planet moves (360°/P)S.

Let us consider the case of an inferior planet, i.e. a planet that will complete one orbit more than Earth before the two return to the same position relative to the Sun.

$\frac{S}{P} 360^\circ = \frac{S}{E} 360^\circ + 360^\circ$

and using algebra we obtain

$P = \frac1{\frac1E + \frac1S}$

For a superior planet one derives likewise:

$P = \frac1{\frac1E - \frac1S}$

Generally, knowing the sidereal period of the other planet and the Earth, P and E, the synodic period can easily be derived:

$S = \frac1{\left|\frac1E-\frac1P\right|}$,

which stands for both an inferior planet or superior planet.

The above formulae are easily understood by considering the angular velocities of the Earth and the object: the object's apparent angular velocity is its true (sidereal) angular velocity minus the Earth's, and the synodic period is then simply a full circle divided by that apparent angular velocity.

Table of synodic periods in the Solar System, relative to Earth:

In the case of a planet's moon, the synodic period usually means the Sun-synodic period. That is to say, the time it takes the moon to run its phases, coming back to the same solar aspect angle for an observer on the planet's surface —the Earth's motion does not affect this value, because an Earth observer is not involved. For example, Deimos' synodic period is 1.2648 days, 0.18% longer than Deimos' sidereal period of 1.2624 d.

## Calculation

### Small body orbiting a central body

In astrodynamics the orbital period $T\,$ (in seconds) of a small body orbiting a central body in a circular or elliptical orbit is:

$T = 2\pi\sqrt{a^3/\mu}$

where:

Note that for all ellipses with a given semi-major axis, the orbital period is the same, regardless of eccentricity.

### Orbital period as a function of central body's density

For the Earth (and any other spherically symmetric body with the same average density) as central body we get

$T = 1.4 \sqrt{(a/R)^3}$

and for a body of water

$T = 3.3 \sqrt{(a/R)^3}$

T in hours, with R the radius of the body.

Thus, as an alternative for using a very small number like G, the strength of universal gravity can be described using some reference material, like water: the orbital period for an orbit just above the surface of a spherical body of water is 3 hours and 18 minutes. Conversely, this can be used as a kind of "universal" unit of time.

For the Sun as central body we simply get

$T = \sqrt{a^3}$

T in years, with a in astronomical units. This is the same as Kepler's Third Law

### Two bodies orbiting each other

In celestial mechanics when both orbiting bodies' masses have to be taken into account the orbital period $P\,$ can be calculated as follows:

$P = 2\pi\sqrt{\frac{a^3}{G \left(M_1 + M_2\right)}}$

where:

• $a\,$ is the sum of the semi-major axes of the ellipses in which the centers of the bodies move, or equivalently, the semi-major axis of the ellipse in which one body moves, in the frame of reference with the other body at the origin (which is equal to their constant separation for circular orbits),
• $M_1\,$ and $M_2\,$ are the masses of the bodies,
• $G\,$ is the gravitational constant.

Note that the orbital period is independent of size: for a scale model it would be the same, when densities are the same (see also Orbit#Scaling in gravity).

In a parabolic or hyperbolic trajectory the motion is not periodic, and the duration of the full trajectory is infinite.

## Earth orbits

orbit center-to-center
distance
altitude above
the Earth's surface
speed period/time in space specific orbital energy
minimum sub-orbital spaceflight (vertical) 6,500 km 100 km 0.0 km/s just reaching space 1.0 MJ/kg
ICBM up to 7,600 km up to 1,200 km 6 to 7 km/s time in space: 25 min 27 MJ/kg
Low Earth orbit 6,600 to 8,400 km 200 to 2,000 km circular orbit: 6.9 to 7.8 km/s
elliptic orbit: 6.5 to 8.2 km/s
89 to 128 min 32.1 to 38.6 MJ/kg
Molniya orbit 6,900 to 46,300 km 500 to 39,900 km 1.5 to 10.0 km/s 11 h 58 min 54.8 MJ/kg
GEO 42,000 km 35,786 km 3.1 km/s 23 h 56 min 57.5 MJ/kg
Orbit of the Moon 363,000 to 406,000 km 357,000 to 399,000 km 0.97 to 1.08 km/s 27.3 days 61.8 MJ/kg

Published - July 2009